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    • CommentRowNumber1.
    • CommentAuthorTodd_Trimble
    • CommentTimeJul 4th 2010

    Added some relevant bits to connected limit, fiber product, and pushout. I wanted to record the result at connected limit that functors preserve connected limits iff they preserve wide pullbacks, which may be a slightly surprising result if one has never seen it before.

    • CommentRowNumber2.
    • CommentAuthorMike Shulman
    • CommentTimeFeb 22nd 2012

    I added a few facts from Bob Pare’s article “Simply connected limits” to connected limit, fiber product, etc., and along the way created L-finite category.

    • CommentRowNumber3.
    • CommentAuthorMike Shulman
    • CommentTimeFeb 22nd 2012

    I added to connected limit a proof that pullbacks and equalizers generate all finite connected limits. I’ve seen it asserted that wide pullbacks and equalizers generate all connected limits, but I don’t see immediately how to generalize this proof to the infinite case. Anyone help?

    I also moved the lemma “wide pullbacks + terminal object => complete” to the page wide pullback (which is actually fiber product).

    I am tempted to split wide pullback off from fiber product; any thoughts? (Actually, I personally think fiber product ought to redirect to pullback, but I’ll defer on that one.)

    • CommentRowNumber4.
    • CommentAuthorMike Shulman
    • CommentTimeFeb 22nd 2012

    Hmm, now I think I see a way to approach the infinite case. Wide pullbacks imply cofiltered limits, and we should be able to construct an arbitrary connected limit as a cofiltered limit of finite connected limits, or at least of L-finite (e.g. finitely generated) connected limits. That makes intuitive sense to me, but it seems like it might be tricky to prove.

    • CommentRowNumber5.
    • CommentAuthorTodd_Trimble
    • CommentTimeFeb 22nd 2012
    • (edited Feb 22nd 2012)

    I agree with splitting off wide pullback. I also support redirecting fiber product.

    • CommentRowNumber6.
    • CommentAuthorMike Shulman
    • CommentTimeMar 1st 2012

    Okay, since no one else expressed an opinion for a week, I turned “fiber product” into wide pullback and redirected “fiber product” to pullback.

    • CommentRowNumber7.
    • CommentAuthorMike Shulman
    • CommentTimeMar 1st 2012

    But the cache bug is operative at fiber product.

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